This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2017.2715561, IEEE Transactions on Power Systems IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. , NO. , 2017
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Sparse State Recovery vs Generalized Maximum-likelihood Estimator of a Power System Junbo Zhao, Student Member, IEEE, Lamine Mili, Fellow, IEEE
Abstract—This letter shows that the sparse state recovery (SSR) optimization method is equivalent to the well-known Huber M-estimator, and then justifies its robustness to bad data. We derive the total influence functions of the Huber M-estimator and the generalized maximum-likelihood (GM)-estimator, and give a formal proof that the Huber M-estimator is vulnerable to bad leverage points while the GM-estimator can handle them. Numerical results carried out on various IEEE systems validate our theoretical results. Index Terms—Bad data, state estimation, sparse state recovery, M-estimator, GM-estimator, influence function, leverage point.
I. I NTRODUCTION ECENTLY, the sparse state recovery (SSR) optimization method proposed for compressed sensing has been advocated for power system static state estimation [1]–[4]. It is claimed that SSR is able to simultaneously suppress bad data (BD) and filter out additive measurement noise by using a mixed 2 - and 1 -norm convex programming. However, no proof was provided to justify that claim. In this letter, we prove that SSR method is equivalent to the well-known Huber M-estimator. We derive the total influence functions of the Huber M-estimator and the generalized maximum-likelihood (GM)-estimator, and give a formal proof that the Huber Mestimator has an unbounded influence function in the presence of bad leverage points while that of the GM-estimator is bounded. In power systems, a leverage point is a power injection measurement on a bus with a relatively large number of incident branches compared to the others (a hub of a graph) or a power injection or a power flow measurement associated with a line having a relatively small reactance compared to the others [5]. II. P ROBLEM F ORMULATION For a power system, the relationship between the measurement vector z˜ ∈ R m and the state vector x ∈ R n can be expressed as ˜ z˜ = h(x) +e ˜, (1) n m ˜ where h(·) : R → R is a vector-valued nonlinear function; e ˜ ∈ Rm is the measurement noise that is supposed to follow a Gaussian distribution with zero mean and covariance matrix R, which consists of the inverse of the measurement variances. Multiplying both sides of (1) by R−1/2 , we obtain the following well-known weighted regression form [ 2]
R
z = h(x) + e,
(2)
where E[eeT ] = I. When there exist BD, the following form is advocated [1]–[4]: z = h(x) + a + e,
(3)
The authors are with the Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Falls Church, VA 22043, USA (e-mail:
[email protected],
[email protected]).
where a is a sparse vector whose element is zero if there is no BD, otherwise it corresponds to BD with arbitrary value. With this model, the goal of the SSR method [1]–[4] is to find x and a that minimize 1 2 J(x, a) = z − h(x) − a2 + λa1 , (4) 2 where λ is the regularization parameter. Note that if λ → ∞, we obtain a = 0 and then (4) reduces to the weighted least squares estimator; by contrast, if λ → 0, (4) reduces to the least absolute value estimator. III. M AIN R ESULTS In this section, we first show that the SSR method is identical to the Huber M-estimator. Then, we derive the total influence functions of the Huber M-estimator and GMestimator, and draw some conclusions. A. Equivalence of the SSR and the Huber M-estimator Theorem 1. The SSR estimator is equal to the Huber Mestimator given by m ρ (ri ), (5) arg min x
i=1
where ri = zi −hi (x) is the ith residual and ρ(·) is the convex Huber cost function expressed as 2 ri /2 for |ri | ≤ λ . (6) ρ (ri ) = λ |ri | − λ2 /2 for |ri | > λ Proof: Rewrite the objective function given by ( 4) as m 1 2 J(x, a) = (ai − ri ) + λ |ai | . (7) 2 i=1 Its necessary and sufficient condition for optimality is ∂J(x, a) ∂J(x, a) = 0 and = 0. ∂a ∂x
(8)
For the first equality, if a i = 0, we have ri = ai (1 +
λ ), |ai |
(9)
which is equivalent to a i = ri − λsign (ri ) if |ri | > λ; otherwise ai = 0, where sign(·) is the signum function of a real number. By substituting a i into (7), we get m 1 2 arg min ri 1|ri |≤λ + λ |ri | − λ2 2 1|ri |>λ (10) 2 x i=1 where 1A : X → {0, 1} is the indicator function of a subset A of a set X. Note that the derivative of (10) with respect to
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2017.2715561, IEEE Transactions on Power Systems IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. , NO. , 2017
TABLE I: Mean Absolute Error of Each Estimator Cases Case 1 Case 2 Case 3
SSR Method 0.0030 0.0985 0.743
Huber M-estimator 0.0031 0.0984 0.7425
GM-estimator 0.0031 0.0402 0.0033
x is equal to that of the second equality condition in ( 8). As a result, (10) is equivalent to (5). Theorem 1 shows the equivalence of the SSR method to the Huber M-estimator that has a bounded influence function of residual, and then justifies its robustness to vertical outliers. The latter refers to outliers that are not in position of leverage [5], [6]. Furthermore, unlike the optimization based solvers, the iteratively reweighted least squares (IRLS) algorithm that is numerical stable and with fast convergence, is advocated to solve the Huber M-estimator. Finally, to have good robustness while being able to achieve high statistical efficiency of the Huber M-estimator under Gaussianity, λ is usually chosen to be between 1 and 3 [5]. B. Influence Functions of M-estimator and GM-estimator To validate the robustness of the SSR/Huber M-estimator to vertical outliers, we derive its total influence function (IF). Note that a bounded IF is the key to justify the robustness of an estimator to outliers. Let the cumulative probability distribution of the residual vector r = z − h(x) be Φ(r), the Huber M-estimator provides an estimate of the state by solving the following implicit equation: m m ∂hi (x) ψ (ri ) = 0, ξi (r, x) = (11) ∂x i=1 i=1 where ψ (ri ) = ∂ρ (ri )/∂ri . By using the definition of the asymptotic total IF and following our previous work [ 7], we derive the IF of the Huber M-estimator as follows: ψ (ri ) T −1 H H ci , (12) IF (r, Φ) = EΦ [ψ (ri )] where ψ (ri ) = ∂ψ (ri )/∂ri ; ci is the ith column vector of HT . By checking the Huber function and its derivative, it is straightforward to verify that if there is no bad leverage point, (12) is bounded. By contrast, if c i is corrupted by gross errors, it is unbounded, yielding unbounded biases of the estimations. To suppress bad leverage points, the GM-estimator is advocated, whose objective function is m i2 ρ(ri /i ), (13) min i=1
where i is the weight to bound the influence of bad leverage point. Following similar steps as before, IF of GM-estimator can be derived as [7] ψ (ri ) T −1 IF (r, Φ) = c i i . (14) H H EΦ [ψ (ri )] Thanks to the weight i , the BD in the position of leverage is bounded as well. Therefore, the GM-estimator is able to suppress both vertical outliers and bad leverage points. To calculate the weight i , Mili, et.al [5] propose to use the projection statistics. Other methods, such as minimum covariance determinant method can also be used. Note that if no outliers occur, GM-estimator reduces to the Huber Mestimator.
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TABLE II: Average Computing Times of Each Estimator Scenarios Case 1 Case 2 Case 3 IEEE 14-bus IEEE 118-bus
SSR Method 0.069s 0.073s 0.075s 0.0303s 0.242s
Huber M-estimator 0.068s 0.074s 0.076s 0.0302s 0.243s
GM-estimator 0.074s 0.082s 0.084s 0.042s 0.32s
IV. S IMULATION R ESULTS To validate the theoretical results, we carry out extensive simulations on IEEE test systems. Specifically, we test the Huber M-estimator, the SSR method [2] and the GM-estimator [5] on the IEEE 30-bus system. λ is set as 1.5 for all three methods; all the tests are performed on a PC with Intel Core i5, 2.50 GHz, 8GB of RAM. The Matlab code can be found at our researchgate website. Three cases are tested: i) no outlier occurs; ii) power injection at Bus 8 is changed from -0.3 p.u to -0.2 p.u to simulate a vertical outlier; (iii) power injection at Bus 2 is changed from 0.18 p.u to 1 p.u to simulate a bad leverage point. The mean absolute error is used as the index to assess the overall performance of each method. Table I shows the mean absolute error of each method for the three cases. It can be found that SSR method has almost the same results as the Huber M-estimator. This is expected as we have proved that SSR method is equivalent to the Huber M-estimator. Furthermore, we find that when vertical outliers occur, the biases of all the estimators increase but remain bounded. This justifies their robustness to vertical outliers. On the other hand, when bad leverage outliers occur, both SSR method and the Huber M-estimator obtain significantly biased estimation results. This verifies our theoretical results that they have unbounded influence functions. By contrast, GMestimator suppresses both vertical outliers and bad leverage points and achieves the highest statistical efficiency. Table II shows the computing times of each method for the three cases as well as the IEEE 14-bus and 118-bus systems without outliers. We observe from this table that SSR method has comparative computing efficiency as the Huber M-estimator. The GM-estimator is the most time consuming approach, but its computing time is in the same order as that of the Huber M-estimator. R EFERENCES [1] W. Xu, M. Wang, J. Cai, A. Tang, “Sparse error correction from nonlinear measurements with applications in bad data detection for power networks,” IEEE Trans. Signal Processing, Vol. 61, No. 24, pp. 6175–6187, 2013. [2] V. Kekatos, G. B. Giannakis, “Distributed robust power system state estimation,” IEEE Trans. Power Syst.,Vol.28,No.2,pp. 1617–1626,2013. [3] W. Zheng, W. Wu, et. al, “Distributed robust bilinear state estimation for power systems with nonlinear measurements,” IEEE Trans. Power Syst., Vol. 32, no. 1, pp. 499-509, 2017. [4] H. Zhu, G. B. Giannakis, “Robust power system state estimation for the nonlinear AC flow model,” in Proceedings of North American Power Symposium (NAPS), 2012. [5] L. Mili, M. Cheniae, N. Vichare, and P. Rousseeuw, “Robust state estimation based on projection statistics,” IEEE Trans. Power Syst., vol. 11, no. 2, pp. 1118–1127, 1996. [6] F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics: The Approach Based on Influence Functions. New York: John Wiley & Sons, Inc., 1986. [7] J. B. Zhao, M. Netto, L. Mili, “A robust iterated extended Kalman filter for power system dynamic state estimation,” IEEE Trans. Power Syst., 2016.
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